In this paper, we study the underlying geometry in the classical Hamilton-Jacobi equation. The proposed formalism is also valid for nonholonomic systems. We first introduce the essential geometric ingredients: a vector bundle, a linear almost Poisson structure and a Hamiltonian function, both on the dual bundle (a Hamiltonian system). From them, it is possible to formulate the Hamilton-Jacobi equation, obtaining as a particular case, the classical theory. The main application in this paper is to nonholonomic mechanical systems. For it, we first construct the linear almost Poisson structure on the dual space of the vector bundle of admissible directions, and then, apply the Hamilton-Jacobi theorem. Another important fact in our paper is the use of the orbit theorem to symplify the Hamilton-Jacobi equation, the introduction of the notion of morphisms preserving the Hamiltonian system; indeed, this concept will be very useful to treat with reduction procedures for systems with symmetries. Several detailed examples are given to illustrate the utility of these new developments.
In this paper we obtain a Hamilton-Jacobi theory for nonholonomic mechanical systems. The results are applied to a large class of nonholonomic mechanical systems, the so-called Čaplygin systems.
Employing a suitable nonlinear Lagrange functional, we derive generalized Hamilton-Jacobi equations for dynamical systems subject to linear velocity constraints. As long as a solution of the generalized Hamilton-Jacobi equation exists, the action is actually minimized (not just extremized).
The geometric formulation of Hamilton--Jacobi theory for systems with nonholonomic constraints is developed, following the ideas of the authors in previous papers. The relation between the solutions of the Hamilton--Jacobi problem with the symplectic structure defined from the Lagrangian function and the constraints is studied. The concept of complete solutions and their relationship with constants of motion, are also studied in detail. Local expressions using quasivelocities are provided. As an example, the nonholonomic free particle is considered.
We discuss an extension of the Hamilton-Jacobi theory to nonholonomic mechanics with a particular interest in its application to exactly integrating the equations of motion. We give an intrinsic proof of a nonholonomic analogue of the Hamilton--Jacobi theorem. Our intrinsic proof clarifies the difference from the conventional Hamilton-Jacobi theory for unconstrained systems. The proof also helps us identify a geometric meaning of the conditions on the solutions of the Hamilton-Jacobi equation that arise from nonholonomic constraints. The major advantage of our result is that it provides us wit…
In this paper, we define Jacobi fields for nonholonomic mechanics using a similar characterization than in Riemannian geometry. We give explicit conditions to find Jacobi fields and finally we find the nonholonomic Jacobi equations in two equivalent ways: the first one, using an appropriate complete lift of the nonholonomic system and, in the second one, using the curvature and torsion of the associated nonholonomic connection.
We extend Hamilton-Jacobi theory to Lagrange-Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton-Jacobi equation as the Dirac-Hamilton-Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the theory specializes to the recently developed nonholonomic Hamilton-Jacobi theory. We are particularly interested in applications to a certain class of degenerate nonholonomic Lagrangian systems with symmet…
In this paper, we consider the hamiltonian formulation of nonholonomic systems with symmetries and study several aspects of the geometry of their reduced almost Poisson brackets, including the integrability of their characteristic distributions. Our starting point is establishing global formulas for the nonholonomic Jacobiators, before and after reduction, which are used to clarify the relationship between reduced nonholonomic brackets and twisted Poisson structures. For certain types of symmetries (generalizing the Chaplygin case), we obtain genuine Poisson structures on the reduced spaces an…
We prove a nonholonomic version of the classical Mauper\-tuis-Jacobi principle which transforms an autonomous mechanical nonholonomic problem, determined by a kinetic minus potential energy and a distribution, in a kinetic nonholonomic problem over a fixed level set of the Lagrangian energy. To prove this result we introduce an appropriate contact bundle structure clarifying the geometric equivalence between both problems. By using the nonholonomic Maupertuis-Jacobi principle, we prove that the regular solutions of a mechanical nonholonomic problem starting from a fixed point and in the same l…
Nonholonomic mechanical systems have been attracting more interest in recent years because of their rich geometric properties and their applications in Engineering. In all generality, we discuss the reduction of a Hamilton-Jacobi theory for systems subject to nonholonomic constraints and that are invariant under the action of a group of symmetries. We consider nonholonomic systems subject to linear or nonlinear constraints, with different positioning with respect to the symmetries. We describe the reduction procedure first, to later reconstruct solutions in the unreduced picture, by starting f…
We develop Hamilton-Jacobi theory for Chaplygin systems, a certain class of nonholonomic mechanical systems with symmetries, using a technique called Hamiltonization, which transforms nonholonomic systems into Hamiltonian systems. We give a geometric account of the Hamiltonization, identify necessary and sufficient conditions for Hamiltonization, and apply the conventional Hamilton-Jacobi theory to the Hamiltonized systems. We show, under a certain sufficient condition for Hamiltonization, that the solutions to the Hamilton-Jacobi equation associated with the Hamiltonized system also solve the…
In this paper we study the problem of Hamiltonization of nonholonomic systems from a geometric point of view. We use gauge transformations by 2-forms (in the sense of Severa and Weinstein [29]) to construct different almost Poisson structures describing the same nonholonomic system. In the presence of symmetries, we observe that these almost Poisson structures, although gauge related, may have fundamentally different properties after reduction, and that brackets that Hamiltonize the problem may be found within this family. We illustrate this framework with the example of rigid bodies with gene…
We introduce a method which allows one to recover the equations of motion of a class of nonholonomic systems by finding instead an unconstrained Hamiltonian system on the full phase space, and to restrict the resulting canonical equations to an appropriate submanifold of phase space. We focus first on the Lagrangian picture of the method and deduce the corresponding Hamiltonian from the Legendre transformation. We illustrate the method with several examples and we discuss its relationship to the Pontryagin maximum principle.
In this paper we study a Hamiltonization procedure for mechanical systems with velocity-depending (nonholonomic) constraints. We first rewrite the nonholonomic equations of motion as Euler-Lagrange equations, with a Lagrangian that follows from rephrasing the issue in terms of the inverse problem of Lagrangian mechanics. Second, the Legendre transformation transforms the Lagrangian in the sought-for Hamiltonian. As an application, we compare some variational integrators for the new Lagrangians with some known nonholonomic integrators.
Non-holonomic mechanical systems can be described by a degenerate almost-Poisson structure (dropping the Jacobi identity) in the constrained space. If enough symmetries transversal to the constraints are present, the system reduces to a nondegenerate almost-Poisson structure on a ``compressed'' space. Here we show, in the simplest non-holonomic systems, that in favorable circumnstances the compressed system is conformally symplectic, although the ``non-compressed'' constrained system never admits a Jacobi structure (in the sense of Marle et al.).
This paper studies hamiltonization of nonholonomic systems using geometric tools. By making use of symmetries and suitable first integrals of the system, we explicitly define a global 2-form for which the gauge transformed nonholonomic bracket gives rise to a new bracket on the reduced space codifying the nonholonomic dynamics and carrying an almost symplectic foliation (determined by the common level sets of the first integrals). In appropriate coordinates, this 2-form is shown to agree with the one previously introduced locally in [34]. We use our coordinate-free viewpoint to study various g…
In this article we develop a theory of contact systems with nonholonomic constraints. We obtain the dynamics from Herglotz's variational principle, by restricting the variations so that they satisfy the nonholonomic constraints. We prove that the nonholonomic dynamics can be obtained as a projection of the unconstrained Hamiltonian vector field. Finally, we construct the nonholonomic bracket, which is an almost Jacobi bracket on the space of observables and provides the nonholonomic dynamics.
The constraint reaction force of ideal nonholonomic constraints in time-dependent mechanics on a configuration bundle $Q\to R$ is obtained. Using the vertical extension of Hamiltonian formalism to the vertical tangent bundle $VQ$ of $Q\to R$, the Hamiltonian of a nonholonomic constrained system is constructed.
We discuss two constructions of the Poisson bivectors based on the Euler-Jacobi theorem for the nonholonomic Stubler model, which describes rolling without sliding of a uniform ball on a cylindrical surface.
In this study we develop a systematic procedure to construct a Poisson operator that describes the dynamics of a three dimensional nonholonomic system. Instead of reducing by symmetry the antisymmetric operator that links the energy gradient to the velocity on the tangent bundle, the system is embedded in a larger space. Here, the extended antisymmetric operator, which preserves the original equations of motion, satisfies the Jacobi identity in a conformal fashion. Thus, a Poisson operator can be obtained by a further time reparametrization. Such Poissonization does not rely on the specific fo…
In this paper we study the reduction of a nonholonomic system by a group of symmetries in two steps. Using the so-called 'vertical-symmetry' condition, we first perform a 'compression' of the nonholonomic system leading to an almost symplectic manifold. Second, we perform an (almost) symplectic reduction, relying on the existence of a momentum map. In this case, we verify that the resulting (almost) symplectic reduced spaces are the leaves of the characteristic foliation of the reduced nonholonomic bracket. On each leaf we study the (Lagrangian) equations of motion, obtaining a nonholonomic ve…
Lagrangian submanifolds are becoming a very essential tool to generalize and geometrically understand results and procedures in the area of mathematical physics. Here we use general Lagrangian submanifolds to provide a geometric version of the Hamilton-Jacobi equation. This interpretation allows us to study some interesting applications of Hamilton-Jacobi equation in holonomic, nonholonomic and time-dependent dynamics from a geometrical point of view.
We show that some modern geometric methods of Hamiltonian dynamics can be directly applied to the nonholonomic Heisenberg type systems. As an example we present characteristic Killing tensors, compatible Poisson brackets, Lax matrices and classical $r$-matrices for the conformally Hamiltonian vector fields obtained in a process of reduction of Hamiltonian vector fields by a nonholonomic constraint associated with the Heisenberg system.
We propose a discretization of vector fields that are Hamiltonian up to multiplication by a positive function on the phase space that may be interpreted as a time reparametrization. We prove that our method is structure preserving in the sense that the discrete flow is interpolated to arbitrary order by the flow of a continuous system possessing the same structure. In particular, our discretization preserves a smooth measure on the phase space to arbitrary order. We present applications to a remarkable class of nonholonomic mechanical systems that allow Hamiltonization. To our best knowledge, …
In this paper, we construct Hamilton-Jacobi equations for a great variety of mechanical systems (nonholonomic systems subjected to linear or affine constraints, dissipative systems subjected to external forces, time-dependent mechanical systems...). We recover all these, in principle, different cases using a unified framework based on skew-symmetric algebroids with a distinguished 1-cocycle. Several examples illustrate the theory.
Based on the d'Alembert-Lagrange-Poincaré variational principle, we formulate general equations of motion for mechanical systems subject to nonlinear nonholonomic constraints, that do not involve Lagrangian undetermined multipliers. We write these equations in a canonical form called the Poincaré-Hamilton equations, and study a version of corresponding Poincaré-Cartan integral invariant which are derived by means of a type of asynchronous variation of the Poincaré variables of the problem that involve the variation of the time. As a consequence, it is shown that the invariance of a certain lin…
In this paper we develop a Hamilton-Jacobi theory in the setting of almost Poisson manifolds. The theory extends the classical Hamilton-Jacobi theory and can be also applied to very general situations including nonholonomic mechanical systems and time dependent systems with external forces.
We elaborate an unified geometric approach to classical mechanics, Riemann-Finsler spaces and gravity theories on Lie algebroids provided with nonlinear connection (N-connection) structure. There are investigated the conditions when the fundamental geometric objects like the anchor, metric and linear connection, almost sympletic and related almost complex structures may be canonically defined by a N-connection induced from a regular Lagrangian (or Hamiltonian), in mechanical models, or by generic off-diagonal metric terms and nonholonomic frames, in gravity theories. Such geometric constructio…
This paper develops the notion of implicit Lagrangian systems on Lie algebroids and a Hamilton--Jacobi theory for this type of system. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of an implicit Lagrangian system on a Lie algebroid $E$ using Dirac structures on the Lie algebroid prolongation $\T^EE^*$. This setting includes degenerate Lagrangian systems with nonholonomic constraints on Lie algebroids.
We consider nonholonomic systems which symmetry groups consist of two subgroups one of which represents rotations about the axis of symmetry. After nonholonomic reduction by another subgroup the corresponding vector fields on partially reduced phase space are linear combinations of the Hamiltonian and symmetry vector fields. The reduction of the Poisson bivectors associated with the Hamiltonian vector fields to canonical form is discussed.
We consider nonholonomic geodesic flows of left-invariant metrics and left-invariant nonintegrable distributions on compact connected Lie groups. The equations of geodesic flows are reduced to the Euler-Poincare-Suslov equations on the corresponding Lie algebras. The Poisson and symplectic structures give raise to various algebraic constructions of the integrable Hamiltonian systems. On the other hand, nonholonomic systems are not Hamiltonian and the integration methods for nonholonomic systems are much less developed. In this paper, using chains of subalgebras, we give constructions that lead…
This paper presents a geometric description of Lagrangian and Hamiltonian systems on Lie affgebroids subject to affine nonholonomic constraints. We define the notion of nonholonomically constrained system, and characterize regularity conditions that guarantee that the dynamics of the system can be obtained as a suitable projection of the unconstrained dynamics. It is shown that one can define an almost aff-Poisson bracket on the constraint AV-bundle, which plays a prominent role in the description of nonholonomic dynamics. Moreover, these developments give a general description of nonholonomic…
This paper develops the theory of Dirac reduction by symmetry for nonholonomic systems on Lie groups with broken symmetry. The reduction is carried out for the Dirac structures, as well as for the associated Lagrange-Dirac and Hamilton-Dirac dynamical systems. This reduction procedure is accompanied by reduction of the associated variational structures on both Lagrangian and Hamiltonian sides. The reduced dynamical systems obtained are called the implicit Euler-Poincaré-Suslov equations with advected parameters and the implicit Lie-Poisson-Suslov equations with advected parameters. The theory …
We discuss the Poisson structures, Lax matrices, $r$-matrices, bi-hamiltonian structures, the variables of separation and other attributes of the modern theory of dynamical systems in application to the integrable Euler top and to the nonholonomic Chaplygin ball.
We generalise Livens theorem, showing that Hamiltonian equation on the vector bundle $E^\ast\rightarrow M$, dual to a general algebroid $E\rightarrow M$, can be derived by means of a variational principle. The framework can be used to describe Hamiltonian dynamics associated with (both vaconomic and nonholonomic) constraints in the bundle $E$.
A close relationship between the classical Hamilton-Jacobi theory and the kinematic reduction of control systems by decoupling vector fields is shown in this paper. The geometric interpretation of this relationship relies on new mathematical techniques for mechanics defined on a skew-symmetric algebroid. This geometric structure allows us to describe in a simplified way the mechanics of nonholonomic systems with both control and external forces.
In this summary of Habilitation Thesis, it is outlined author's 18 years research activity on mathematical physics, geometric methods in particle physics and gravity, modifications and applications (after defending his PhD thesis in 1994). Ten most relevant publications are structured conventionally into three "strategic directions": 1) nonholonomic geometric flows evolutions and exact solutions for Ricci solitons and field equations in (modified) gravity theories; 2) geometric methods in quantization of models with nonlinear dynamics and anisotropic field interactions; 3) (non) commutative ge…
In this paper, we propose a discrete Hamilton--Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton--Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton--Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton--Jacobi equation…
We develop a Hamilton-Jacobi theory for singular lagrangian systems in the Skinner-Rusk formalism. Comparisons with the Hamilton-Jacobi problem in the lagrangian and hamiltonian settings are discussed.
In this paper, we derive a "hamiltonian formalism" for a wide class of mechanical systems, including classical hamiltonian systems, nonholonomic systems, some classes of servomechanism... This construction strongly relies in the geometry characterizing the different systems. In particular, we obtain that the class of the so-called algebroids covers a great variety of mechanical systems. Finally, as the main result, a hamiltonian symplectic realization of systems defined on algebroids is obtained.
In this paper we propose a geometric Hamilton--Jacobi theory on a Nambu--Jacobi manifold. The advantange of a geometric Hamilton--Jacobi theory is that if a Hamiltonian vector field $X_H$ can be projected into a configuration manifold by means of a one-form $dW$, then the integral curves of the projected vector field $X_H^{dW}$ can be transformed into integral curves of the vector field $X_H$ provided that $W$ is a solution of the Hamilton--Jacobi equation. This procedure allows us to reduce the dynamics to a lower dimensional manifold in which we integrate the motion. On the other hand, the i…
In this paper, some of formulations of Hamilton-Jacobi equations for Hamiltonian system on Lie algebroids are given. Here we use the general properties of Lie algebroids to express and prove two geometric version of the Hamilton-Jacobi theorem for Hamiltonian system on Lie algebroids. Then this results are generalized and two types of time-dependent Hamilton-Jacobi theorem of Hamiltonian system on Lie algebroids are obtained.
The aim of this paper is to develop a Hamilton--Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton-Jacobi equation accordingly to the Hamiltonian and the evolution vector fields for a given Hamiltonian function. We also analyze the corresponding formulation on the symplectification of the contact Hamiltonian system, and establish the relations between these two approaches. In the last section, some examples are discussed.
Recently, M. de León el al. ([8]) have developed a geometrical description of Hamilton-Jacobi theory for multisymplectic field theory. In our paper we analyse in the same spirit a special kind of field theories which are gauge field theories. The Hamilton-Jacobi theory for this kind of fields is shown.
In this paper we extend the geometric formalism of the Hamilton-Jacobi theory for time dependent Mechanics to the case of classical field theories in the k-cosymplectic framework.
Recently, M. de León el al. ([9]) have developed a geometric Hamilton-Jacobi theory for Classical Field Theories in the setting of multisymplectic geometry. Our purpose in the current paper is to establish the corresponding Hamilton-Jacobi theory in the Cauchy data space, and relate both approaches.
In this paper we extend the geometric formalism of the Hamilton-Jacobi theory for hamiltonian mechanics to the case of classical field theories in the framework of multisymplectic geometry and Ehresmann connections.
We develop a Hamilton-Jacobi theory for singular lagrangian systems using the Gotay-Nester-Hinds constraint algorithm. The procedure works even if the system has secondary constraints.
In nonholonomic mechanics, the presence of constraints in the velocities breaks the well-under\-stood link between symmetries and first integrals of holonomic systems, expressed in Noether's Theorem. However there is a known special class of first integrals of nonholonomic systems generated by vector fields tangent to the group orbits, called {\it horizontal gauge momenta}, that suggest that some version of this link should still hold. In this paper we prove that, under certain conditions on the symmetry Lie group, the (nonholonomic) momentum map is conserved along the nonholonomic dynamics, t…
